Stability of twisted states in the Kuramoto model on Cayley and random graphs

TL;DR

This paper investigates how the structure of different quasirandom graphs affects the stability and behavior of solutions in the Kuramoto model, revealing both similarities and differences in dynamics across graph types.

Contribution

It provides new insights into the stability of twisted states and the relation between network structure and dynamics in the Kuramoto model on Cayley and random graphs.

Findings

Twisted states are stable on complete and Paley graphs but not on others.

Solutions on complete and random graphs stay close over finite times with similar initial conditions.

New results on synchronization and stability of twisted states in the Kuramoto model.

Abstract

The Kuramoto model (KM) of coupled phase oscillators on complete, Paley, and Erdos-Renyi (ER) graphs is analyzed in this work. As quasirandom graphs, the complete, Paley, and ER graphs share many structural properties. For instance, they exhibit the same asymptotics of the edge distributions, homomorphism densities, graph spectra, and have constant graph limits. Nonetheless, we show that the asymptotic behavior of solutions in the KM on these graphs can be qualitatively different. Specifically, we identify twisted states, steady state solutions of the KM on complete and Paley graphs, which are stable for one family of graphs but not for the other. On the other hand, we show that the solutions of the IVPs for the KM on complete and random graphs remain close on finite time intervals, provided they start from close initial conditions and the graphs are sufficiently large. Therefore, the results of this paper elucidate the relation between the network structure and dynamics in coupled nonlinear dynamical systems. Furthermore, we present new results on synchronization and stability of twisted states for the KM on Cayley and random graphs.